Mixed finite element methods for viscoelastic flow analysis: a review

Mixed finite element methods for viscoelastic flow analysis: a review1

Frank P.T. Baaijens*

Dutch Polymer Institute, Eindhoven University of Technology, Faculty of Mechanical Engineering,

P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

Received 2 April 1998; revised 14 May 1998

Abstract

The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow

problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting

with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS

and the so-called EVSS-G method are discussed among others. Furthermore, stabilization techniques such as the streamline

upwind Petrov±Galerkin (SUPG) and the discontinuous Galerkin (DG) are reviewed. The performance of the numerical

schemes for both smooth and non-smooth benchmark problems is discussed. Finally, the capabilities of viscoelastic flow

solvers to predict experimental observations are reviewed. # 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

During the last decade, significant progress has been made in the development of numericalalgorithms for the stable and accurate solution of viscoelastic flow problems. In particular, for a numberof benchmark problems for steady flows, agreement between a number of different formulations hasbeen demonstrated at ever increasing values of the Weissenberg number, see Refs. [1,2]. And, moreimportantly, mesh convergent results are achieved without lowering the maximum achievableWeissenberg number. However, limits in the maximum attainable Weissenberg number still exist.

This work aims to review the progress made during, aproximately, the period 1987±1997. Reviewscovering work prior to this may be found in the book of Crochet et al. [3] and the review of Keunings[4]. A restriction is made to mixed finite element methods to solve viscoelastic flows using constitutiveequations of the differential type. A variety of alternative formulations have been developed during thelast decade as well, for instance, streamline integration methods, finite volume methods, and spectralcollocation methods; but these are not considered in detail in this work although some references are

J. Non-Newtonian Fluid Mech. 79 (1998) 361±385

ÐÐÐÐ

* Corresponding author.1Dedicated to Professor Marcel J. Crochet on the occasion of his 60th birthday.

0377-0257/98/$ ± see front matter # 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 0 2 5 7 ( 9 8 ) 0 0 1 2 2 - 0

made for comparison purposes. It is believed that prominent and measurable improvements have beenmade for mixed finite element methods.

To discuss the algorithmic developments within the framework of closed-form differentialconstitutive equations it is sufficient to, initially, restrict attention to the Oldroyd-B model. Theessential features that separates viscoelastic flow calculations from, for instance, generalizedNewtonian problems are present in this model, and the algorithmic developments can be wellillustrated using this model.

To start with, the governing set of equations is recalled. After a brief review of the key numericalstrategies to resolve the viscoelastic flow problem, attention is focussed on mixed finite elementmethods in Section 4. Specific features of the mixed formulation for steady flows are discussed indetail. Unsteady viscoelastic flows may either be solved by simply introducing a time-stepping schemein the steady flow solver, but also enables the use of specific algorithms, which are discussed inSection 5. In Section 6 solution strategies are discussed to cope with the large number of unknowns thatgenerally results after discretization. In Section 7 the performance of several schemes for flows throughsmooth and non smooth geometries is evaluated. Also, the ability of several algorithm to analyze thestability of viscoelastic flows is reviewed. Finally, in Section 8, the predictive capabilities of theviscoelastic flow solver is compared with experimental results.

2. Governing set of equations

The analysis of viscoelastic flows involves the solution of a coupled set of partial differentialequations: the equations representing the conservation of mass, momentum and energy, and constitutiveequations for a number of physical quantities present in the conservation equation such as density,internal energy, heat flux, stress, etc.

In this work, only incompressible, isothermal viscoelastic flows of homogeneous materials areconsidered with specific choice of the type of constitutive equations. It is believed that such flows areprototypical and of sufficient generality to discuss the development of numerical methods specific forviscoelastic flows. For instance the introduction of non-isothermal effects does not appear to introducespecific difficulties that require special algorithmic developments other than already included informulations for isothermal flows [5].

For incompressible, isothermal flows the conservation equations for mass and momentum may beexpressed as, respectively

r � u � 0; (1)

�Du

Dt� � @u

@t� u � ru

� �� ÿrp�r � T � f ; (2)

where � denotes, in this case, the constant density, D=Dt the substantial or material time derivative, pthe pressure, T the extra stress tensor and f a general volumetric force (expressed here as force per unitvolume). Frequently, the extra stress tensor is defined in terms of a viscous and a viscoelasticcontribution:

T � 2�eD� s (3)

362 F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 79 (1998) 361±385

where �e denotes an effective viscosity and D the symmetric part of the velocity gradient:

D � 12�ru�ruT� (4)

with T denoting the transpose, and s denoting the viscoelastic contribution. Note that in Eq. (3) theeffective viscosity can represent the viscosity of the solvent (for a polymer solution) or part of the total(apparent) stress viscosity of the system according to the model employed for the stress. The splittingexpressed in Eq. (3) is arbitrary and user-defined, but, as we will see further below, it can havesignificant repercussions on the stability of the numerical scheme.

As mentioned before, the set of Eqs. (1) and (2) is incomplete without the specification of the extrastress tensor s. A large variety of approaches exists to define this extra stress tensor. Roughly, adistinction can be made between closed form constitutive models of the integral and differential type(e.g. [6]), and models based on, for instance, kinetic or molecular dynamics theories (e.g. [7]) that donot yield a closed form constitutive model. In this work, only constitutive models of the differentialtype are considered having the following structure:

� sr�sÿ 2�D� g�s� � 0 (5)

where� represents a characteristic relaxation time for the polymer system, � the polymer viscosity and g�s�a non-linear function of the stress which goes to zero at least as fast as quadratically as s approaches zero. Anumber of well-known constitutive models are represented by Eq. (5), including the upper-convectedMaxell, FENE-P, Phan-Thien±Tanner and Giesekus model. Note that in general, both � and � can beconsidered as functions of the stress or other state variables, but since this does not seem to influence thebehavior of the various numerical schemes to be discussed below, they will be assumed constants here forsimplicity. In Eq. (5) the symbolr denotes the upper-convected time derivative defined as

sr � @s

@t� u � rsÿ �ru�T � sÿ s � ru (6)

Clearly, Eq. (5) may not describe the actual mechanical behaviour of many viscoelastic fluids withsufficient accuracy. For instance, it is well accepted that even in the limit of small deformations a spectrumof relaxation times is necessary to accurately describe the rheology of most viscoelastic fluids. Such aspectrum frequently may be represented by a finite set of independent relaxation times. To each relaxationtime a constitutive equations of the form of 5 may be used. However, to discuss the algorithmicdevelopments, a single relaxation time with a constitutive equation that obeys Eq. (5) is sufficient.

3. Approaches to viscoelastic finite element computations

To illustrate the typical difficulty encountered in the numerical analysis of viscoelastic flows,consider the most simple representation of the models given by Eq. (5), the upper-convected Maxwell(UCM) model. Assuming an Eulerian formulation, which is the most intuitive for flow problems, theone-relaxation time model may be written as

�@s

@t� u � rsÿ �ru�T � sÿ s � ru

� �� sÿ 2�D � 0 (7)

F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 79 (1998) 361±385 363

As a consequence of the convective term u � rs, the constitutive equation is a partial differentialequation rather than an ordinary differential equation as would be the case in a Lagrangian approach.

Two fundamentally different approaches to this issue can be discerned: firstly, a mixed formulationmay be adopted in which besides the velocity u, and the pressure p, the extra stress tensor s i/s treatedas an additional unknown and each of the equations, hence including the constitutive equation, ismultiplied independently with a weighting function and transformed in a weighted residual form (see,for instance, [3] for a comprehensive review of early methods in the category).

Secondly, the constitutive equation may be transformed into an ordinary differential equation (ODE).For transient problems this can, for instance, be achieved in a natural manner by adopting a Lagrangianformulation [8]. For many applications, however, this leads to excessive mesh distortions and frequentmesh updating is necessary. Alternatively, for steady flows within an Eulerian framework theconstitutive equation may be integrated along streamlines, e.g. [9±12]. Neither of the two latterformulation require a separate discretization of the extra stress tensor. For unsteady flows, an operatorsplitting method may be applied in which the convective part is split from the remainder of theconstitutive equation.

A number of constitutive equations of the integral type cannot be transformed into a differentialequation. For such models either a Lagrangian or a streamline integration formulation must be adopted.

In this review, only mixed formulations within the Eulerian framework are discussed.

4. Mixed finite element formulations

Consider the steady, incompressible, inertialess flow of an Oldroyd-B fluid. As a point of departurethe classical three-field mixed formulation is chosen, in which, besides the momentum and continuityequation, the constitutive equation is also cast in a weighted residuals form. This is a natural extensionof the common velocity±pressure formulation for Stokes type problems and implicitly accounts for thepartial differential form of the constitutive equation.

Problem 1 (MIX). Find s; u and p such that for all admissible weighting functions S; v and q

�S; � sr�sÿ 2�D� � 0 (8)

��rv�T; 2�eD� s� ÿ �r � v; p� � 0; (9)

�q;r � u� � 0 (10)

where (.,.) denotes the appropriate inner product. Most of the early work on viscoelastic flow analysis isbased on this formulation, see, for example, [13±16]. If during discretization the pairs fs; Sg, fu; vgand fp; qg are chosen from the same spaces, this approach is referred to as the Galerkin formulation.Two basic problems exist with the above formulation: (i) with increasing value of the Weissenbergnumber, the importance of the convective term u � rs grows and a Galerkin discretization as applied inEq. (8) is not optimal [17], (ii) the discretization spaces for the three variables need to be carefullyselected with respect to each other to satisfy the so-called LBB (Ladyzhenskaya±Babuska±Brezzi) orinf±sup condition, see [18]. The evolution of several approaches to resolve these problems arediscussed below.


4.1. Convective term

The most widespread method to account for the convective term in the constitutive equation is the so-called streamline±upwind/Petrov± Galerkin (SUPG) method of Brooks and Hughes [19], first applied toviscoelastic flows by Marchal and Crohet [20]:

�S� �u � rS; � sr�sÿ 2�D� � 0: (11)

Over the years several variations of the parameter � have been proposed, but are all of the form

� � h

U; (12)

where h is a characteristic length-scale of the element and U a characteristic velocity. Reported choicesfor the scaling velocity U include the velocity in the direction of the local coordinates at the midpoint ofa bi-quadratic element [20], or at each integration point [21], the norm of the velocity u, [17], or acharacteristic velocity of the flow, [22].

At steep stress boundary layers or near singularities the SUPG method may produce oscillatory stressfields [20,23]. To circumvent this, Marchal and Crochet [20] proposed a streamline-upwind formulation(SU), where the upwind term �u � rS is applied to the convective term of the constitutive equationonly:

�S; � sr�sÿ 2�D� � ��u � rS; u � rs� � 0: (13)

For a number of benchmark problems (e.g. stick-slip, 4-to-1 contraction) convergence up to veryhigh values of the Weissenberg number may be obtained with this formulation. However, apart from thefact that this formulation is inconsistent substitution of the exact solution � s

r �sÿ 2�D � 0 leaves thesecond term of Eq. (13) as a residual, the method is only first-order accurate with respect to the extrastresses, as demonstrated by Crochet and Legat [24].

An alternative to the SUPG or SU method is the discontinuous Galerkin (DG) or Lesaint±Raviartmethod. Here the extra stress tensor is approximated discontinuously from one element to the next, andupwind stabilization is obtained as follows:

�S; � sr�sÿ 2�D� ÿ

XN

e�1

Zÿ in

e

S : u � n�sÿ sext� dÿ � 0; (14)

with n the unit outward normal on the boundary of element e, ÿ ine the part of the boundary of element e

where u � n < 0, and sext the extra stress tensor in the neighboring upwind element. In the context ofviscoelastic flows, this method was first introduced by Fortin and Fortin [25] based on ideas of Lesaintand Raviart [26] who proposed the method to solve the neutron transport equation. Compared to theSUPG formulaion, the implementation of the discontinuous Galerkin method in a standard finiteelement code is much more involved. This is due to the boundary integral along the inflow boundary ofeach element where stress information of the neighboring, upwind, element is needed. This drawback iscircumvented for unsteady flows by Baaijens [27] by using an implicit/explicit implementation.


Saratimo and Piau [28] have applied a combination of the discontinuous Galerkin method forconstant interpolated stress components, and the Baba±Tabata method for the bi-linear interpolatedstresses. Under these circumstances the above advection schemes are monotonic.

Yet another alternative is provided by Singh and Leal [29] who apply a third-order upwind scheme ofTabata and Fujima [30], while special measures were taken to maintain a positive definite conformationtensor.

4.2. LBB condition

In the well known velocity±pressure formulation of the Stokes problem, recovered by omitting s fromEqs. (9) and (10) a compatibility condition between the velocity and pressure interpolation needs to besatisfied. This compability condition is known as the LBB-condition, [18]. Likewise the addition of theweak form of the constitutive equation, Eq. (8), imposes compatibility constraints on the interpolationof the triple stress±velocity±pressure. Fortin and Pierre [31] have shown that in the absence of a purelyviscous contribution, e.g. �e�0 and using a regular Lagrangian interpolation, the following conditionsmust hold: (i) The velocity±pressure interpolation must satisfy the usual LBB condition to prevent forlocking and spurious oscillation phenomena, (ii) if a discontinuous interpolation of the extra stresstensor s is used, the space of the strain rate tensor D as obtained after differentiation of the velocity fieldu must be a member of the interpolation space of the extera stress tensor s, i.e. D � s, and (iii) if acontinuous interpolation of s is used, the number of internal nodes must be larger than the numberof nodes on the side of an element used for the velocity interpolation. This confirms the earlier workof Marchal and Crochet [20] who introduced a 4-by-4 bi-linear subdivision of the extra stresses on abi-quadratic velocity element. Condition (ii) is relatively easily satisfied using a discontinuous Galerkinmethod, see [25,27,32,33].

Baranger and Sandri [34,35] have shown that the third condition does not need to be imposed if apurely viscous contribution is present (�e 6�0), which allows a much larger class of discretizationschemes.

This result emphasizes the need to retain an elliptic contribution of the form ��rv�T;D� in theweak form of the momentum equation, Eq. (9). One way to achieve this in the absence of apurely viscous contribution is the application of a change of variables, known as the elastic viscousstress splitting (EVSS) formulation, first introduced by Perera and Walters [3] and Mendelsonet al. [36] for the flow of a second-order fluid and later extended to viscoelastic flows by Beris et al.[37,38]:

R � sÿ 2�D (15)

Substitution of this into Eqs. (8) and (9), respectively, yields

�S; �Rr�R� 2��D

r� � 0; (16)

ÿ��ru�T; 2�� e�D� s� � �r � u; p� � 0: (17)

Unfortunately, the change of variables (Eq. (15)) does not yield a closed expression for everyconstitutive equation. Furthermore, the convected derivative of the rate of strain tensor emerges, which


requires a second-order derivative of the velocity field. Two approaches are known to circumvent this:either an integration by parts is performed on the weak formulation (which necessitates the introductionof additional boundary conditions on D), or D is considered as a separate unknown and is obtained byan L2-projection of the velocity gradient:

�E; 2Dÿ �ru� �ru�T�� 0 (18)

where E denotes a suitable weighting function. This approach, due to Rajagopalan et al. [39], is usedby many others, i.e. [32,40] [41±54]. Sun et al. [50] recently proposed an adaptive EVSS method wherethe viscosity in the change of variables in Eq. (15) is adapted to retain a sufficient amount of ellipticityin the momentum equation:

R � sÿ 2�aD; �a � hjj�ijjjmax

jjuijjmax

max

; (19)

with h a characteristic element length, and ui and � ij are components of the velocity vector and extrastress tensor, respectively.

One may proceed one step further by performing a projection of �E; Gÿ �ru�T� � 0, and usingG as an additional unknown rather than D, and subsequently use the projection of G in the constitutiveequations as well. The latter leads to the so-called EVSS-G method, introduced by Brown et al. [55]and Szady et al. [56].

Recently, Guenette and Fortin [47] have introduced a modification of the EVSS formulation, knownas the discrete EVSS method (DEVSS). In this method a stabilizing elliptic operator is introduced in thediscrete version of momentum equation. This is similar to the EVSS method but the objective derivativeof the rate of strain tensor is avoided and the method is not restricted to a particular class of constitutiveequations. Using the UCM model and SUPG, this formulation reads:

Problem 2 (DEVSS/SUPG). Find �D; s; u; p such that for all admissible weighting functions

E; S; v and q

�S� �u � rS; � sr�sÿ 2�D� � 0; (20)

��ru�T; 2��Dÿ �D� � s� ÿ �r � v; p� � 0; (21)

�q;r � u� � 0; (22)

�E;Dÿ �D� � 0: (23)

In the discrete momentum Eq. (21), an elliptic operator 2��Dÿ �D� is introduced, where �D is adiscrete approximation of the rate-of-strain tensor D obtained from Eq. (23). If the exact solution isrecovered, this elliptic operator vanishes. However, in a finite element calculation this is generally notthe case. Baaijens et al. [53] have first applied DEVSS in combination with the discontinuous Galerkinmethod.

In analogy with the EVSS-G method, the DEVSS-G method may be defined where a projection ofthe velocity gradient �ru�T is made instead of the strain rate tensor. This projection is also introducedin the weak form of the constitutive Eq. (20):


Problem 3 (DEVSS-G/SUPG). Find G; s; u; p such that for all admissible weighting functionsE; S; v and q:

S� �u � rS; u � rsÿ G � sÿ s � Gÿ ��G� GT�ÿ � � 0; (24)

��rv�T ; 2��Dÿ 12�G� GT�� s� ÿ �r � v; p� � 0; (25)

�q;r � u� � 0; (26)

�E; �ru�T ÿ G� � 0: (27)

The DEVSS-G-based methods (either in combination with SUPG, DG or other upwind schemes ofsufficient accuracy), are expected to supersede the EVSS±G-based formulations, because of the abovecited advantages. The additional cost and complexity of the use of the discrete approximation of thevelocity gradient tensor in the weak form of the constitutive equation as in Eq. (24) is marginal incomparison with the DEVSS method, Eq. (20). However, for steady flows the use of the DEVSS-G/DGformulation was not found to result in an enhanced performance compared to the DEVSS/DGformulation by Baaijens et al. [53]. In fact, the DEVSS-G-based methods appear to be mostadvantageous during stability analysis of viscoelastic flows.

4.3. Spatial discretization

With respect to the velocity-pressure discretization the usual requirement imposed by the LBBcondition need to be respected.

Prior to the introduction of (D)EVSS, is was customary to use an equal order interpolation of thevelocity and stress field. Exceptions to this are the so-called 4�4 element of Marchal and Crochet [20],and discontinuous interpolations of the extra stresses by Fortin and Fortin [25]. Due to the introductionof (D)EVSS, a larger selection is possible, yet the most common scheme is to use a stress and strain ratediscretization that is one order lower than the velocity interpolation.

For the EVSS formulation this also apears to hold if higher order interpolations are applied, see[44,48,57,58]. When using the basic mixed formulation, MIX, and higher order (spectral) elements,results of van Kemenade and Deville [59] and [60] and Warichet and Legat [58] indicate that forsmooth flows best performance is obtained with a polynomial space of the extra stresses that is oneorder higher than for the velocity field.

If (D)EVSS(-G) and a continuous interpolation of the extra stress tensor are used, the strain ratetensor (or the velocity gradient tensor) is interpolated in the same way as the extra stress tensor. Themost commonly applied element for the (D)EVSS(-G) method has a bi-quadratic velocity, bi-linearpressure, stress and strain rate (or velocity gradient) interpolation, see [42,55,61].

In case of the discontinuous Galerkin method, hence with a discontinuous interpolation of the extrastress tensor, a variety of choices have been experimented with by Fortin et al. [25,32,62] and Baaijens[27,33,53]. The most recent implementation [53], using DEVSS, shows that with a bi-quadraticvelocity interpolation and a bi-linear discontinuous interpolation of the stresses, the most stable resultsare obtained with a bi-linear continuous interpolation of the strain rate tensor. This differs from theDEVSS/SUPG formulation where the rate of strain tensor is interpolated equal to the extra stresstensor.


Saratimo et al. [28,63,64] have applied the so-called Raviart-Thomas element where the diagonalcomponents of the extra stress tensor are interpolated discontinuous piece-wise constant while the off-diagonal components are interpolated with a continuous bi-linear polynomial. Clearly, the piece-wiseconstant interpolation limits the accuracy to first order only.

A special class of methods if formed by so-called (pseudo-) spectral collocation formulations, seee.g. [65±67]. These may generally be viewed as a special form of MIX and EVSS although thediscretization is not of the Galerkin type since the weighting function are chosen to be unity at thecollocation points. Different types of interpolation functions (e.g. Chebyshev or Legendre polynomialsand Fourier sin and cos functions) may be used for different spatial directions, while combination withfinite difference approximations is also reported, e.g. Beris and coworkers [65,68,69]. Initially, spectralcollocation methods were only applied to relatively simple domains, but this restriction is remedied bythe application of domain decomposition techniques, Souvaliotis and Beris [7].

5. Time-dependent flows

Probably the most direct extension to unsteady flows of the mixed formulations discussed above arebased on an implicit temporal discretization as used by Northey et al. [71], Brown et al. [55] Bodart andCrochet [72] and [73], and Szady et al. [56] resulting in a fully coupled set of equations. Baaijens [21]used a time discontinuous Galerkin least squares formulation for the temporal integration.

By introducing a selective implicit/explicit treatment of various parts of the equations, a certaindecoupling at each time step of the set of equations may be achieved to improve computationalefficiency. For instance Singh and Leal [29] first applied the three-step operator splitting methoddeveloped by Glowinksi and Pironneau [74] to viscoelastic flows, later followed by Saratimo et al.[28,63] and [64] and Luo [52]. In the first and third step a generalized Stokes problem is obtained,while in the second step a convection±diffusion type of problem needs to be solved. This offers thepossibility to apply dedicated solvers to sub-problems of each fractional time step.

The implicit/explicit Newton-like implementation of the discontinuous Galerkin method by Baaijens[27] allows the elimination of the extra stresses on the element level at each time step. The resulting setof equations has the size of a Stokes problem in the regular velocity-pressure setting.

Carew et al. [75±77] have developed a Taylor±Petrov±Galerkin algorithm that also decouples the setof equations at each fractional time step.

Avgousti et al. [78] and Beris and Surshkumar [79] apply a time splitting/influence matrix methodsin combination with a spectral collocation method to study three-dimensional unsteady viscoelasticflows.

Besides the three step �-method previously cited, Glowinski and Pironneau [74] also describe a two-step operator splitting method forming the basis of a number of ODE-type methods for time dependentviscoelastic flows. Essentially, this operator splitting method is a method of characteristics where attime t � tn�1, the material derivative Ds=Dt � @s=@t � u � rs is approximated by

Ds

Dt� s�x; tn�1� ÿ s�p; tn�

tn�1 ÿ tn

(28)

where p denotes the position at time tn of the particle located at x at time tn�1. Several alternatives havebeen suggested to obtain the tensor s�p; tn�. Either a discrete particle is traced back in time and space or


a pure convection problem is solved, like in Fortin and Esselaoui [80], Basombrio et al. [81] andKabanemi et al. [82] who all use a weak form of the method of characteristics, while Baaijens [83]applied a time-discontinuous Galerkin least squares method.

In the Lagrangian approach followed by Rasmussen and Hassager [8] the constitutive equation istrivially transformed into an ODE, but frequent remeshing is generally needed to assure the quality ofthe mesh and hence the solution and interpolation are introduced upon passing information tosubsequent meshes. Harlen et al. [84] have developed a split Lagrangian±Eulerian formulation wherestress variables are associated with the nodes that follow the flow. Remeshing is performed by aDelaunay triangulation algorithm with a constant number of nodes, thereby avoiding interpolation fromone mesh to the other, as stresses are associated with nodes.

Notice that rather than solving the steady flow problem as such, one may use a time-marchingprocedure to approach steady-state as a limiting case. In particular, the splitting of the set of equationsas is acheived in the �-scheme is of interest to reduce memory requirements as each of the sub problemsis significantly smaller than the full coupled set of equations.

6. Solution technology

The resulting set of non-linear equations is often solved using a Newton±Raphson scheme inconjunction with a first order continuation in the Weissenberg number Wi. The resulting linear set ofequations is generally solved using a direct LU factorization and a frontal solver, [20,39]. This leads towhat is usually refered to as a fully coupled approach; the full set of equations is solved simultaneously.

However, mixed methods typically result in a large number of unknowns. Even for two-dimensionalproblems with modest geometrical complexity and a single relaxation time memory limitations prohibitthe use of direct solvers. One reason for this is the need for fine meshes to resolve the steep stressboundary layers near curved boundary and singularities that occur with many of the existingconstitutive equations. Consequently, the coupled approached requires a (frequently too) large amountof memory and may lead to excessive CPU time consumption. In particular, for three dimensionalcomputations, the coupled approach with direct solvers appears not feasible.

One way to reduce memory requirements is by solving the viscoelastic extra-stresses separately formthe momentum and continuity equation. Coupling between the two sets of equations is achievediteratively by means of Picard iteration. However, convergence of this scheme is slow and the attainableWeissenberg number is usually significantly lower than with a coupled solver. By application of apreconditioned GMRES iterative solver to enforce coupling, Fortin et al. [32,40,62], substantiallyincreased the limiting Weissenberg number and reduced the number of iterations required to reachconvergence, while retaining a decoupled solution procedure. the procedure has some similarity withthe three-step �-method of Glowinski and Pironneau [74] for unsteady flows. Baaijens [85] has alsoapplied a GMRES iterative solver using the discontinuous Galerkin method, but has designed a specialpreconditioner tailored to specific features of this method.

Methods to solve the linear set of algebraic equations resulting from the fully-coupled Newton'smethod by preconditioned, preferably matrix-free, iterative solvers like GMRES and BICGSTAB havereceived little attention yet, but need exploration to efficiently and robustly solve three-dimensionalsteady and unsteady viscoelastic flow problems. An exception is the work of Tsai and Liu [86] whoinvestigated the performance of the BiCGSTAB and GMRES iterative solvers in conjuction with an


incomplete LU factorization as preconditioner. Computations are based on the SU formulation ofMarchal and Crochet [20]. Although a CPU-time reduction upto a factor of 2 is reported in a number oftest problems, a tight drop tolerance on the incomplete LU factorization has to be applied leading to alarge memory consumption.

Keunings [87] reviewed the application of a parallel finite element algorithms for the solution ofviscoelastic flow problems. In particular, the application to streamline integration based formulationsare investigated, but application to mixed formulation appears feasible, such as reported by Zone et al.[88]. Domain decomposition based techniques (see, for example, [89]) appear to be particularlyattractive.

7. Performance evaluation

7.1. Flow with smooth solutions

During the past decade a number of benchmark problems with smooth geometries have beenproposed and analysed: the journal bearing problem (e.g. [37±39,56,90]), flow through an undulatedtube (e.g. [24,44,56,57,65,68,70,91±94]) and the flow along an array of cylinders (e.g.[44,46,54,57,70,93,95]).

However, the falling sphere in a tube problem is by far the most cited. Although a variety ofconstitutive models have been used only results for the upper convected Maxwell model will be citedhere. The benchmark problem is defined as follows. The sphere with radius R is located at thecenterline of the tube with radius Rc. The tube wall moves parallel to the centerline with a velocity V inthe positive z-direction. The ratio of the cylinder radius Rc and the sphere radius R is � � Rc=R � 2.The Weissenberg number is defined as

Wi � �V

R: (29)

The drag F0 on a sphere falling in an unbounded Newtonian medium is given by

F0 � 6��RV : (30)

It is customary to compare the so-called drag correction factor given

K�Wi� � F�De�F0

; (31)

where F is the drag on the cylinder as a function of the Weissenberg number.Table 1 gives an overview of currently available results, with the restriction that only reference is

made to studies that report drag correction factor beyond a Weissenberg number of 2, while results onthe finest mesh available are included. These results have also been depicted graphically in Fig. 1.Previously [1], a number of studies appeared to indicate a limiting value of the Weissenberg number of1.6 that seemed to point to the existence of a physical limit point [55]. However, careful meshrefinement to capture the exceptionally steep stress boundary layers along and at the wake of the spherehas now allowed several studies to proceed beyond a Weissenberg number of 2. However, significant


discrepancy exists between various methods beyond this point. The results of the EEME method of Jinet al. [96], the adaptive hp method of Warichet and Legat [58] (which is similar to the method reportedby Yurun and Crochet [48]), and the DEVSS/DG of Baaijens et al. [53] agree remarkably well upto aWeissenberg number of 2.2±2.4. The results of the OS/SUPG method of Luo [52] and Sun et al. [50]

Table 1

Comparison of computed drag correction factor for the falling sphere problem. Only those results have been included that

exceed a Wi number of 2 and are obtained on the finest mesh reported

Methods/Ref. EEME/[96] OS/SUPG/[52] Adaptive hp [58] DEVSS/DG/[53] AVSS/SI/[50] EVSS/SUPG/[45]

Wimax 2.2 2.8 2.45 2.7 3.2 2.2

Wi

0.4 5.1827 5.1894 5.1862 5.186 5.2222

0.8 4.5277 4.5249 4.5274 4.528 4.5355

1.0 4.3415 4.3326 4.3405 4.341 4.3290 4.3354

1.4 4.1273 4.1075 4.1335 4.136 4.0957

1.8 4.0409 3.8851 4.0557 4.058 3.9504

2.0 4.0377 3.7403 4.0453 4.049 3.9387 4.0492

2.2 4.0623 3.6637 4.0475 4.052 3.8934 4.0786

2.4 3.5924 4.0580 4.065 3.8791

2.6 3.5242 4.087 3.8961

2.8 3.4326 3.8864

3.0 3.8889

3.2 3.9306

Fig. 1. Graphical comparison of computed drag correction factor for the falling sphere problem. Only those results have been

included that exceed a De number of 2 and are obtained on the finest mesh reported.


significantly deviate from these, most likely caused by an insufficient mesh refinement. AlthoughBaaijens [33] obtained results for a Weissenberg number beyond 3 using a first-order discontinuousGalerkin method having similar solution characteristics as the 4�4 SU method of Crochet and Legat[24], convergence with mesh refinement could not be established.

In summary, convergent results up to reasonably high values of the Weissenberg number (largerthan 2) have been obtained for the falling-sphere-in-a-tube benchmark problem using a variety ofmethods, including EVSS, DEVSS and EEME-based methods, as a well as hp methods, provided thatsufficiently refined meshes have been used.

It is unclear at this point whether results at even higher values of the Weissenberg number can beobtained with second or higher order methods when more refined meshes are used, although this mayindeed be expected based on previous experience. Nevertheless, it is fair to state that significantprogress has been made over the past decade.

7.2. Flows with geometric sigularities

Viscoelastic flows with geometric singularities, like sharp corners in contraction or expansion flowsand stick-slip transitions, have proven to be notoriously difficult to solve. This holds in particular forthe upper-convected Maxwell and Oldroyd model, while for other constitutive models like Phan-Thien±Tanner, Giesekus, Leonov, FENE (-p) etc., quite reasonable results have been obtained: See Section 8on comparison with experiments.

In three recent papers, Davies and Devlin [97], Hinch [98] and Renardy [99], sought local solutionsnear the singular point. Renardy assumed a Newtonian velocity field near the singular point andintegrated the Maxwell model numerically exact along the streamlines. Renardy reported boundary-layer-like regions near the upstream and downstream walls with a separable stress of order rÿ0.91.Davies and Devlin, and Hinch, on the other hand, found analytical solutions for the Oldroyd modelassuming a separable form of the stress field. Hinch showed that in the core region the stresses are ofthe order rÿ2/3 and the strain rate of order rÿ4/9, while at the upstream wall the stresses are again oforder rÿ2/3 but the strain rate is of order rÿ1/3. Davies and Devlin found solutions that include Renardy'sand Hinch's results and one with a stress field of order rÿ0.985.

Consequently, the (semi-) analytical work cited above is, unfortunately, inconclusive.For the flow of an Oldroyd-B fluid through an axisymmetric 4-to-1 contraction and using the 4�4 SU

formulation, Marchal and Crochet [20] achieved solutions upto a Weissenberg number beyond 60. Inthis case, the Weissenberg number is taken as the relaxation time times the fully developed wall shearrate at the downstream channel. Similar results are reported for the stick-slip problem. However, con-vergence with mesh refinement has not been demonstrated and in view of Crochet and Legat [24] theaccuracy of these results may be questioned. Similar remarks hold for the results of low-order constantstress interpolation in conjunction with the discontinuous Galerkin method reported by Baaijens [33].

As shown by Keunings [100], the maximum attainable Weissenberg number decreases withincreasing mesh resolution. This finding is confirmed by Apelian et al. [101] for the stick-slip problememploying the EEME/SUPG formulation and by Coates et al. [102] for the flow through anaxisymmetric 4-to-1 contraction. In fact, Coates et al. showed that the corner singularity almost behavesas rÿ1, while the order of the singularity (rÿl) should satisfy �<1 for the solution to be square integrableto apply the SUPG method. Singh and Leal [103] argued that standard discretizations insufficientlycapture the �-dependence of the stress field. They have obtained mesh converged solutions provided


sufficient �-resolution is applied. Their numerical algorithm is based on an operator splitting methodwith third-order upwinding and a specific strategy is included to preserve positive definiteness of theconformation tensor. Using a finite volume method, Sasmal [104] reached solutions upto a Weissenbergnumber in excess of 24, but convergence with mesh-refinement has not been demonstrated. In relationto this, Salomon et al. [105] have shown the difficulty of handling singular points even when using aNewtonian fluid.

Baaijens [85] demonstrates that using the DEVSS/DG method, stable and accurate results can beobtained upto Weissenberg numbers beyond 24 as well, while the limiting Weissenberg numberincreases with continued mesh refinement. Moreover, the theoretical results of Hinch [98] with respectto the order of the corner singularity for the axisymmetric 4-to-1 contraction are recovered numerically.Furthermore, comparison of computed corner vortex intensities with results obtained by the finitevolume method of Sasmal [104] and the EEME/SUPG results of Coates et al. [102], indicates the loworder convergence behaviour of the discretization chosen by Sasmal.

Results of other recently proposed methods like DEVSS/SUPG have not been reported yet for theseflows using UCM or Oldroyd-B fluids. Thus far, spectral methods do not appear to resolve the cornersingularity problem robustly, as demonstrated by Yurun and Marchal [48].

In summary, fully mesh converged results for the axisymmetric 4-to-1 contraction problem using theUCM or the Oldroyd-B model have not been obtained despite the convergence with mesh refinementfor the majority of the flow domain using the DEVSS/DG method [85]. Convergence with meshrefinement, however, could not be established along the downstream wall.

7.3. Stability analysis of complex flows

In recent years a growing interest in the analysis of purely elastic instabilities can be observed (see[106] for a recent review on this topic as well as references therein, [78,92,107±115].

The computational analysis of the stability of viscoelastic flows has proven to be a major challenge.This is amply demonstrated by Brown et al. [55] and Szady et al. [56], who computed the linearstability of a planar Couette flow of the upper-convected Maxwell to access the numerical stability of anumber of mixed finite element formulations. Theoretical results have shown that this inertialess flowis stable for any value of the Weissenberg number. Methods based on a Galerkin, EEME and EVSSformulation in combination with either SUPG or SU demonstrated a limiting Weissenberg beyondwhich the numerical solution became unstable. It is for this reason that the EVSS-G/SUPG and EVSS-G/SU formulations were introduced for which no limiting Weissenberg number was found within therange of Weissenberg numbers and mesh resolutions examined.

Both Northey et al. [71] (EEME/SUPG) and Bodart and Crochet [72] (4�4 SUPG and SU)investigated the stability of the Taylor±Couette flow by direct numerical integration of the unsteadyequations. The capability of the 4�4 SUPG and SU to correctly predict the stability of the planarCouette flow is not reported.

8. Comparison with experiments

With the improved performance of numerical methods for viscoelastic flow simulations, directcomparison of the numerical with the experiment results becomes increasingly feasible and necessary.


Comparison may be based on global flow features such as settling velocity of particles, drag ofobstacles, die-swell and pressure-drop, or on locally measured flow kinematics by streak-linephotography, laser Doppler velocimetry (LDV) or particle imaging velocimetry (PIV), and even on(locally) measured normal stresses by employing the stress optical rule. Apart from purely numericalissues (e.g. convergence properties), the predictive capabilities of any numerical analysis are only asgood as the input data: Constitutive models, material parameters and boundary conditions. Con-sequently, comparison of experimental and numerical results provides detailed information on theperformance of constitutive models in complex flows rather than in viscometric flows only and mayprovide guidelines for improvements thereof.

Fortunately, for many transparent polymeric liquids both polymer stresses and flow kinematics canbe measured. For instance by application of LDV or (digital) PIV it is possible to map the full three-dimensional velocity field in complicated geometries. The stress-optical rule is frequently applied toconvert flow induced birefringence (FIB) data to normal and shear stresses. For polymer melts this isusually achieved by means of crossed polarizers providing field-wise informations in the form of fringepatterns that are correlated to a norm of the shear stress and first normal stress difference. (see e.g.[116±120]). Measurement of both birefringence and extinction angle allows a separation of shear andnormal stresses. For polymer solutions, on the other hand, crossed polarizers generally cannot be useddue to the low stresses and stress-optical coefficient providing insuficient birefringence signal. In thatcase the rheo-optical analyser (ROA) developed by Fuller and Mikkelsen [121] may be appplied givinga point-wise measurement of flow birefringence. The application to polymer melts is limited becauseseveral order transitions may occur within the diameter of the laser beam (usually of order 100 mm).

Stress-optical measurements are generally applied to two-dimensional planar flows althoughextensions to axisymmetric ([122]) and three-dimensional flows ([123]) are currently underinvestigation. However, purely two-dimensional flows do not exist experimentally and edge effectsmay disturb the direct comparison of computational and experimental results. Furthermore, the stress-optical rule may not hold which indeed is the case at finite extensions and near the glass transitiontemperature for polymer melts.

8.1. Solutions

As illustrated in the book of Boger and Walters [124], streakline photography is widely applied forflow visualization. Townsend and Walters [125] investigated the expansion flow in two- and three-dimensional geometries of a Newtonian liquid and several polymer solutions. Good agreement betweenpredicted and measured flow patterns is reported. The viscoelastic analysis is based on a SUPG/Taylor±Galerkin algorithm using a single mode PTT model. Similarly, Debbaut et al. [126,127] and laterPurnode and Crochet [128], examined the flow of polyacrylamide solutions through planarcontractions. In [128] it is demonstrated that with a single mode FENE-P model, available rheologicaldata (both in shear and extension) could be described satisfactorily, and flow patterns could bepredicted, albeit at different flow rates.

Rajagopalan et al. [129] investigated the flow of a polystyrene solution in an eccentric cylinderconfiguration and compared point-wise measured stresses with predictions based on a 4-mode Giesekusmodel. For most of the flow domain good quantitative agreement is reported. Davidson et al. [130,131]utilized a polystyrene solution in a smooth planar periodically constricted channel. Both velocity andstresses are measured point-wise using LDV and FIB. In [131] measured results are primarily compared


with Newtonian predictions. The use of a single-mode UCM and White±Metzner model proved to givea qualitative agreement for normal stresses for the latter.

In a sequence of papers Quinzai et al. [132±134] have subsequently investigated the rheologicalproperties of a 5 wt.% polyisobutylene in tetradecane (PIB/C14) solution in relation to a number ofexisting constitutive models and have measured velocity and stress distributions for the flow through aplanar contraction. In [133] velocity and normal stresses at a number of cross sections are measured andcompared with predictions of Newtonian and power-law fluids. The order of the singularity near thereentrant corner is demonstrated to (nearly) obey a similarity form that holds for power-law fluids.Measured axial velocity profiles along the symmetry line in a planar 4-to-1 contraction are used in[134] to access the capabilities of a number of constitutive models to predict the transient elongationalstresses. Of the models investigated, the multi-mode PTT model performed best. Fair agreement withnumerical results of these experimental findings is reported by Baaijens [83] using a multi-mode PTTmodel in conjuction with an operator splitting approach, Mitsoulis [135] using a K-BKZ integral modeland a streamline integration-based numerical method and Azaiez et al. [51,136] using single and multi-mode versions of the PTT and Giesekus model.

To study the flow around a confined cylinder, Baaijens et al. [137,138], apply the same PIB/C14solution as Quinzani et al. [132]. Velocities and stresses are indentified by means of LDV and FIB,respectively. Calculations based on a discontinuous Galerkin method are in quantitative agreement withexperimental results, provided a multi-mode linear PTT model is applied. Alternatively, the multi-modeGiesekus model, in particular, overestimated the normal stresses at the wake of the cylinder where aplanar elongational flow exists. Similar results are reported in [139] using an integral-type K-BKZmodel.

For axisymmetric flows, Li and Burghardt [122] and Li et al. [140], show that rather thandecomposing optical signals into stress data, optical properties can be used for comparison purposes.Particularly, simple expressions are obtained in the limit of small retardation and the methodology isdemonstrated for a Newtonian fluid in [122] and for viscoelastic fluids in [140]. The attractiveness ofthe axisymmetric stagnation flows studied in this work is that both uni- and bi-axial elongationalproperties of the fluid may be examined.

Following Becker et al. [141], Arigo et al. [45] and Rajagopalan et al. [49] investigated the steadyand unsteady motion of a falling sphere in a tube filled with a Boger fluid. At steady flow LDV is usedto access the velocity along the centerline, while at transient flow conditions digital particle imagingvelocimetry (DPIV) is utilized. No signs of flow instabilities are observed. To describe the motion ofthe sphere accurately, it is demonstrated that a multi-mode version of the linear PTT model, with non-linearity parameters chosen to capture the elongational data, is needed. The attainable Weissenbergnumber is limited by the inability to resolve stress boundary layers near the wall of the sphere and at thewake of the sphere. Computations are based on the EVSS-formulation. Satrape and Crochet [41]compare computed drag on a falling sphere using a FENE model with experimental data of Chhabraet al. [142] (see [143] for related experimental results).

8.2. Melts

A significant portion of the work on numerical analysis of viscoelastic flow of polymer melts is basedon streamline integration methods employing KBKZ-type constitutive models with a damping functiondue to Papanastasiou, Scriven and Macosko [144] (PSM) or Wagner [145].


For example, Goublomme and Crochet [11,146] analysed the extrudate swell of high-densitypolythylene (HDPE) melt as reported by Koopmans [147] using a KBKZ model, see also related workin [148,149]. With a modified Wagner damping function giving a non-zero second normal stressdifference quantitative agreement with experimental results could be obtained, albeit at a large ratio ofsecond and first normal stress difference. Furthermore, it was shown that non-isothermal effects hadlittle effect on the swelling ratio.

Park et al. [150] also applied the KBKZ model to analyse the flow of a HDPE melt through a planarcontraction. Good agreement between measurement and predicted normal stresses along the centerlineis reported. Ahmed and Mackley [151,152] measured both velocities and birefringence along thecenterline of a planar contraction for two grades of HDPE melts. The analysis is based on a KBKZmodel with a Wagner type damping function. With material parameters based on shear data only, aquantitative agreement with experimental data could be achieved for one of the grades only (Natene).The form of the damping function did not give independent control over shear and elongational datanecessary to describe the behavior of the other grade (Rigidex). Recently, Beraudo et al. [153] have alsoreported results on contraction flows of LDPE melts and made comparison with the experimentallyobserved birefringence patterns.

Similarly, Kiriakidis et al. [154,155] investigated the contraction flow of a LLDPE melt, also with theKBKZ model and a PSM damping function. Both a short and a long exit die length are examined. Inparticular, for the short exit length good quantitative agreement between predicted and measurednormal stresses along the centerline is found. The same flow configuration is analysed by Maders at al.[156] using a single-mode White±Metzner model.

Hulsen and van der Zanden [10] applied an eight-mode Giesekus model to the analysis of contractionflows of an LDPE melt, obtaining a quantitative agreement with the experimentally observed vortexsizes, similar to Dupont and Crochet [157] and Luo and Mitsoulis [148].

Kajiware et al. [158] simulated the converging flow of a LDPE through a tapered slit die using asingle-mode PTT and Giesekus model. In the case of the PTT model, computed and predicted normaland shear stresses are reported to deviate only 20%.

Baaijens et al. [53] investigated the flow of a LDPE melt around a confined cylinder, also reported byHartt and Baird [159]. The numerical analysis is based on an implicit/explicit implementation of thecombination of the DEVSS and discontinuous Galerkin method allowing the efficient handling ofmultiple relaxation times. Parameters of the exponential version of the PTT model and the Giesekusmodel are fitted to available shear data including viscosity and first normal stress difference. For thePTT model no unique set of parameters could be identified without the presence of elongationaldata. Three different parameter sets have been applied, all giving an equally good fit of the sheardata. Comparison with experimentally obtained birefringence patterns revealed that neither of themodels could fully quantitatively predict the observed stress patterns, consistent with results reportedin [159].

9. Conclusions

Over the past decade, significant progress has been made in the numerical and experimental analysisof viscoelastic flows. Within the category of mixed methods the so-called DEVSS-based methods, asfirst introduced by Guenette and Fortin [47], appear to provide the most robust formulations currently


available. To achieve accurate results, the DEVSS method should preferably be combined with eitherthe SUPG formulation [56], the DG formulation [53] or other upwind schemes that are of sufficientaccuracy such as the third order accurate upwind technique of Singh and Leal [29] or Taylor±Galerkinmodel-based methods [77]. The use of streamline upwind (SU) techniques or low-order interpolation incombination with the DG method yields over-diffusive algorithms that are too inaccurate for practicalcalculations despite their robustness.

For the analysis of the stability of viscoelastic flows, best results are reported by Szady et al. [56],using the EVSS-G/SUPG formulation. Equivalent results are obtained using the DEVSS-G/SUPGmethod.

For time-dependent flows the operator splitting methodology developed for the Navier±Stokesequations by Glowinsky and Pironneau [160], known as the �-scheme, is gaining popularity. It providesfor an elegant splitting of the momentum and continuity equation on the one hand, and the consti-tutive equation on the other. As a consequence, a sequence of well defined problems of moderatesize need to be computed which are easily applicable to problems with multiple relaxation times.The �-scheme has been sucessfully applied to a variety of weak formulations, including DEVSS-basedmethods.

In the analysis of flows through smooth domains, such as the falling-sphere-in-a-tube benchmarkproblem, convergent results have been obtained upto high values of the Weissenberg number with avariety of numerical methods. However, limits in the attainable Weissenberg number still exist. It is atpresent unclear what causes these limits, but it is expected that convergence to higher values of theWeissenberg number may be achieved with increasing mesh refinement, for instance, because in thatcase stress boundary layers can be represented with more accuracy.

For flows with geometrical singularities, however, still significant problems exist. On a given mesh,SUPG-based methods, in particular, appear to provide convergent results upto disappointingly smallWeissenberg numbers only. Moreover, convergence of the iterative scheme (Newton±Raphson)deteriorates with increasing mesh refinement. Using appropriate discretization, relatively high values ofthe Weissenberg number are achieved using the DEVSS/DG method for the axisymmetric 4-to-1problem using the UCM and Oldroyd-B fluid. Yet, mesh convergent results near the singularity andalong the downstream wall could not be obtained. But, in contrast with the DEVSS/SUPG method, everhigher Weissenberg numbers are achieved with increased mesh refinement.

Confrontation with experimental results increasingly reveals the inability of existing constitutivemodels to predict the complicated stress fields in viscoelastic flows. This holds in particular for regionswith strong elongational flows. Consequently, improved constitutive models are desired. Directconfrontation of computational and experimental results may provide guidelines for such models.

A relatively recent and promising approach that does not require closed form constitutive models arethe so-called micro±macro formulations based on kinetic theories, as first suggested by Laso andOÈ ttinger [161] and Feigl et al. [162], and later adopted and improved by Hua and Schieber [163], Bellet al. [164], Laso et al. [165], Hulsen et al. [166], OÈ ttinger et al. [167] and Halin et al. [168].

Acknowledgements

Both Antony Beris and Roland Keunings have been of great help in preparing annotated referencesand I would like to thank them for making this information available to me.


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Mixed finite element methods for viscoelastic flow analysis: a review